163 



successive values of the stages of the continuous fraction: 



1 



2 +J 



1 + 1 



1 + 1 



1 + etc., 



- values which oscillate alternately towards the positive or negative 

 with respect to a definite limiting number, to which the successive 

 terms continuously approach more closely. This true limit-va]ue 

 is no other than the irrational number: |(3 1/5), which represents 

 the smaller portion of the ratio known as the "aurea sectio", - 

 a ratio which since the days of Leonardo da Vinci (1452 1519) 

 has been considered to be intimately connected with all questions 

 about ideal visual beauty of proportion in art and natural forms x ). 

 The "ideal" arrangement in phyllotaxis, towards a "tendency" 

 in living nature appears to exist, should therefore be considered such, 

 that a spiral arrangement is attended to, whose characteristic angular 

 divergence is equal to 7r(3 1/5), i. e. to 137 30'28". In this case 

 true "orthostichies" do no longer exist, because there can never 

 be a leaf standing exactly above some other, except in infinity. In 

 the opinion of the adherents of this theory, the "ideal" disposition of 

 leaves about a cylindrical stem aimed at by nature, would, therefore, 

 be such as to prevent each leaf from overlapping another, even if 

 the plants were so closely packed together as is often the case in dark 

 tropical forests. The question, in how far this teleological view must 

 be considered as being a mere fiction, or in real agreement with 

 the natural adaptation of the plant to its need of light and free air, 

 may be passed over here 2 ). 



30. If the theory of phyllotaxis just explained be once adopted, 



1 ) Let a straight line AB be equal to unity, and C be a point so situated on 

 it, that AC : CB = AB : AC. Then AC 2 = AB. BC, from which follows that 

 BC = 4(3 1/5), and AC = (J/5 1). This division of AB by the point C 

 is called the "golden section", "aurea sectio" (also: "sectio divina" or "divina 



proportio" (Kepler)), the length of^both portions is 0,381988 and 0,618034 



respectively. The relations of this ratio to the properties of the regular pentagon 

 and, therefore, to pentagonal symmetry in general, (are wellknown.J 



2 ) J. Wiesner, Flora, (1875), p. 115, 139, 142; Biol. Centralblatt., 23, 209, 

 249, (1913); H. Winckler, Pringsheim's Jahrbuch f. wiss. Botan., 36, 1, (1901). 

 Wiesner concludes: "Regular phyllotaxis as determined in the sense described 

 above, is a phenomenon doubtless intimately connected with the question of the 

 most suitable adaptation to the natural conditions of light-absorption by plants". 



